Methods and guidelines for effective model calibration

Convergence Criteria
Convergence criteria are needed for the modified Gauss-Newton iterative process. In 

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UCODE and MODFLOWP, parameter estimation converges if either one of two convergence cri-
teria are satisfied. First, convergence is achieved when the largest absolute value of d
j
r
/b
j
r
, j=1,NP, 
is less than a user-defined convergence criterion (TOL of UCODE and MODFLOWP). That is,
|d
j
r
/b
j
r
| < TOL for all j=1,NP
(7)
where d
j
r
is the jth element of d
r
, the parameter change vector of equation 4; b
j
r
is the ith element 
of b
r,
the vector of parameter values being changed in equation 4; and NP is the number of estimat-
ed parameters. If b
j
r
equals 0.0, 1.0 is used in the denominator. Preferably, this convergence crite-
rion is satisfied by the final calibrated model with TOL assigned a value no larger that 0.01.
Second, the nonlinear regression converges if the sum of squared objective function (eq. 1 
or 2) changes less than a user-defined amount (SOSR of UCODE and MODFLOWP) for three se-
quential iterations. This convergence criteria often is useful early in the calibration process to avoid 
lengthy simulations that fail to improve model fit.
Log-Transformed Parameters
The parameters in vector b of equation 1 can either be the native values directly relevant to 
the system being considered, or the log-transform of the native values. Log-transforming parame-
ters can produce an inverse problem that converges more easily, and prevents the actual parameter 
values from becoming negative (Carrera and Neuman, 1986). In UCODE and MODFLOWP, the 
log-transform is implemented using the natural logarithm, but the input and output include base 10 
logarithms because these are easier for most modelers to use (MODFLOWP was converted to the 
base 10 user interface in version 3.3).
UCODE and MODFLOWP are designed so that even when there are log-transformed pa-
rameter values, the user generally sees the more readily understood native values. Thus, for exam-
ple, even when parameters are log-transformed, the starting parameter values specified by the user 
are native values. There are, however, three situations in which either model input or output are 
affected by a parameter being log-transformed.
The one model input situation occurs when there is prior information on the log-trans-
formed parameter value, in which case there can only be one parameter included in the prior infor-
mation (one term in the summation presented after eq. 1), and the specified statistic needs to be 
related to the base 10 log of the parameter. The statistic can be calculated using methods described 
under Guideline 6, described later in this report.

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The first model output situation is fairly subtle and will not be noticed by most users. It 
involves calculation of the damping parameter and the convergence criteria of equation 4, which 
are calculated to control or measure the change in the native parameter values. The printed damp-
ing parameter value, therefore, can not always be derived easily by the user. Calculation of the 
damping parameter is described in Appendix B.
The second model output situation is that log-transformed parameter estimates, standard 
deviations, coefficients of variation, and confidence interval limits appear in the output file along 
with the exponential of these values. In most circumstances, the log-transformed values are ignored 
by the user and the native values are used instead. Related issues are discussed in the section "Stan-
dard Deviations, Linear Confidence Intervals, and Coefficients of Variation" later in this report.